In an article entitled "Razionalita della Involuzioni Piane," Castelnuovo proved the rationality of all involutions in the plane. In his notation, a plane involution is defined by four equations between (α, β, γ), the points of a plane, and (x1, x2, x3, x4), the points on a surface in space of three dimensions:
ρxn = fn(α, β, γ) (n = 1, 2, 3, 4),
the functions fn being rational forms of the same degree. To a given point in the plane corresponds uniquely a point of the surface; but to a given point on the surface correspond, in general, not one point, but several points, in the plane. When to a point on the surface corresponds a group of two or more points in the plane, these groups are said to constitute an involution in the plane. From the point of view of the plane itself, an involution means the establishment of an ∞2 series of point groups, each point of the plane belonging to one and only one group, and each point determining the remainder of the points associated wit it to form a group. The important theorem that all such involutions are rational signifies that a one-on-one correspondence may be established between their groups and the points of a plane. hence, if the points in the above equations range over a surface I, each point of I giving rise to a group of n points in the plane π, it is always possible to establish a mutually unique correspondence between the points of I and the points of a plane π'. Thus every involution may originate in a plane π' so that to each point of π' corresponds exactly one group of the involution in π. The problem may then be transferred from the study of surfaces to the study of systems of curves in the plane, as follows.
On account of the rationality of the surface, the above equations may be replaced by relations connecting the points (ξ η, ζ) of π' with the points (x,y,z) of π:
ρξ = U(x,y,z),
ρη = V(x,y,z),
ρζ = W(x,y,z),
or ξ:η:ζ = U:V:W.
To a point (ξ1, η1, ζ1) corresponds in π a group of points determined as the variable intersections of U/ξ1 = V/η1 = W/ζ1, i.e., the intersections outside of the fundamental points common to U, V, and W.
Thus every involution may be generated by a linear net of curves
λU + μV + νW = 0.
The involution will be characterized as an In, n being the number of points in a group, or the number of intersections of two curves of the net outside of the base points; n will be call the "order" of the involution.
There is thus established a general theory of involutions of any order and any kind. In this theory, as formulated by Castelnuovo, reference is made to the so-called "cyclic" involutions, i.e., those in which the transformation is effected by a periodic birational transformation, either a collineation or a periodic Cremona transformation. The theory of the cyclic periodic transformations of any order has been developed by Kantor and Wiman. The involutions of order 2 were first classified by Bertini, these, as is known, being entirely cyclic. There has apparently as yet been no attempt to distinguish between the geometric construction of the cyclic and the non-cyclic types, a distinction which would rise for the first time in the case of the I3. It is the purpose of this paper ti discuss the construction of an I3 from the point of view of nets of curves, and to differentiate, in the case of nets of deficiency 0, 1, and 2, between those which lead to cyclic and to non-cyclic involutions. for the systems of curves of the above deficiencies the cyclic cases will be found, and in each case the analytic expression of the periodic transformation set up.